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Properties

Label 2-4400-1.1-c1-0-82
Degree 22
Conductor 44004400
Sign −1-1
Analytic cond. 35.134135.1341
Root an. cond. 5.927405.92740
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·7-s + 9-s + 11-s − 4·17-s − 4·19-s − 4·21-s + 6·23-s − 4·27-s + 2·29-s − 8·31-s + 2·33-s − 4·37-s − 6·41-s + 6·43-s + 2·47-s − 3·49-s − 8·51-s − 12·53-s − 8·57-s − 4·59-s + 14·61-s − 2·63-s − 10·67-s + 12·69-s − 8·71-s + 4·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.970·17-s − 0.917·19-s − 0.872·21-s + 1.25·23-s − 0.769·27-s + 0.371·29-s − 1.43·31-s + 0.348·33-s − 0.657·37-s − 0.937·41-s + 0.914·43-s + 0.291·47-s − 3/7·49-s − 1.12·51-s − 1.64·53-s − 1.05·57-s − 0.520·59-s + 1.79·61-s − 0.251·63-s − 1.22·67-s + 1.44·69-s − 0.949·71-s + 0.468·73-s + ⋯

Functional equation

Λ(s)=(4400s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4400s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 44004400    =    24⋅52⋅112^{4} \cdot 5^{2} \cdot 11
Sign: −1-1
Analytic conductor: 35.134135.1341
Root analytic conductor: 5.927405.92740
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4400, ( :1/2), −1)(2,\ 4400,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
11 1−T 1 - T
good3 1−2T+pT2 1 - 2 T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+2T+pT2 1 + 2 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1+8T+pT2 1 + 8 T + p T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.191871686050505723023772771752, −7.22844652283952877540616710847, −6.72548837346794712093746847989, −5.91079469741736939188176872846, −4.88770977827577808579170180890, −3.98331514047171746360978742031, −3.28810060334468870197516710928, −2.57127588045710357001575296894, −1.67005665302555494693104159770, 0, 1.67005665302555494693104159770, 2.57127588045710357001575296894, 3.28810060334468870197516710928, 3.98331514047171746360978742031, 4.88770977827577808579170180890, 5.91079469741736939188176872846, 6.72548837346794712093746847989, 7.22844652283952877540616710847, 8.191871686050505723023772771752

Graph of the ZZ-function along the critical line